A284391 1-limiting word of the morphism 0 -> 1, 1 -> 001.

1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1

Offset

1

Comments

The morphism 0 -> 1, 1 -> 001 has two limiting words. If the number of iterations is even, the 0-word evolves from 0 -> 1 -> 001 -> 11001 -> 00100111001; if the number of iterations is odd, the 1-word evolves from 0 -> 1 -> 001 -> 11001 -> 00100111001 -> 110011100100100111001, as in A284391. The 0-limiting word results from the 1 limiting word by replacing the initial 00 by 1.

Conjecture: the limiting frequency of 0's in both limiting words is 1/2.

From Adnan Baysal, Aug 06 2023: (Start)

Conjecture: Excluding the first term, if runs of 1s of length one are replaced by 0, and runs of 1s of length three are replaced by 1, we get the same sequence.

Conjecture: Index distance between two consecutive 1s is either one or three. Excluding the first term, if distances of one are replaced by 0, and distances of three are replaced by 1, we get the same sequence. (End)

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Mathematica

s = Nest[Flatten[# /. {0 -> {1}, 1 -> {0, 0, 1}}] &, {0}, 9]; (* this sequence *)
Flatten[Position[s, 0]];  (* A284392 *)
Flatten[Position[s, 1]];  (* A284393 *)

Crossrefs

A284388 shifted right.

Cf. A003159, A284392, A284393.

Sequence in context: A267869 A068434 A323511 * A127015 A281114 A286749

Adjacent sequences: A284388 A284389 A284390 * A284392 A284393 A284394

Keyword

nonn,easy

Author

Clark Kimberling, Mar 30 2017

Status

approved